A u ? v matrix A with entries from Q is image partition regular provided that, whenever N is finitely colored, there is some ~x 2 Nv with all entries of A~x lying in one color. Image partition regular matrices are natural tools for representing some classical theorems of Ramsey Theory, including theorems of Hilbert, Schur, and van der Waerden. Several characterizations and consequences of image partition regularity were investigated in the literature. Many natural analogues of known characterizations of image partition regularity of finite matrices with rational entries over the integers have been generalized for matrices with entries from reals over the ring (R, +). In both the cases of reals and integers, usual ordering played an important role. In the present work we shall prove that natural analogues of known characterizations of image partition regularity of finite matrices with rational entries over the integers are also valid for matrices with entries from Gaussian rationals Q[i] over the ring of Gaussian integers Z[i]. The main hurdle for this generalization is the absence of ordering, and to overcome this hurdle we need some modifications of established techniques. We also prove that Milliken-Taylor Matrices with entries from Z[i] are also image partition regular over Z[i].
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